Non-integer linear array doa estimation method based on accelerated iterative hard thresholding

Abstract

The non-integer linear array DOA estimation method based on accelerated iterative hard threshold value comprises the following steps: constructing a model comprising a non-integer linear array and received signals thereof; generating an estimated value of an ideal covariance matrix of the received signal model; constructing a difference co-array and a sparse optimization model of the non-integer linear array according to the received signal model; solving the sparse optimization model by using an iterative hard threshold algorithm combined with an Armijo type step length criterion to automatically adjust the step length and a support set, and obtaining an initial angle estimation; and generating an accurate angle estimation by performing a first-order Taylor expansion on the initial angle estimation. The application expands the array degrees of freedom by constructing a difference co-array, takes a single fast sample covariance vector as a sparse representation input, adaptively accelerates the convergence by combining the iterative hard threshold algorithm and the Armijo step length criterion, and compensates for the off-grid error caused by grid division by using a Taylor expansion total least square method, so that high-precision DOA estimation is realized under a small number of fast shots.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing and direction of arrival estimation, and particularly relates to a high-precision DOA estimation method applicable to non-integer linear arrays. Background Technology

[0002] Direction of Arrival (DOA) estimation, a core technology in array signal processing, infers the angle information of the incident wave by analyzing the received signal from the array. It plays a crucial role in systems such as radar detection, wireless communication, and sonar positioning. Its estimation accuracy and signal resolution are mainly affected by the array geometry and the number of available data snapshots.

[0003] Traditional high-resolution algorithms are mostly based on uniform linear array structures, such as the MUSIC and ESPRIT algorithms, which utilize the Vandermonde property to perform eigenvalue decomposition on the sample covariance matrix, thereby separating the signal and noise subspaces. To overcome the limitation of the number of physical array elements on the number of estimable sources, researchers have proposed sparse array configurations such as nested arrays and coprime arrays. By constructing differential co-arrays, larger virtual apertures are obtained, and spatial smoothing techniques are combined to adapt to subspace-based algorithms. Another technical approach is sparse reconstruction methods based on compressed sensing, which discretize the continuous angle domain to form an overcomplete dictionary, transforming DOA estimation into a sparse signal recovery problem. Representative methods include those based on... Convex optimization algorithms based on norms and sparse Bayesian learning were developed. Furthermore, strategies such as phase difference projection and two-step offset correction were developed to address phase ambiguity and off-grid effects for non-uniform or arbitrary array configurations.

[0004] However, these methods still face significant challenges when applied to non-integer linear arrays. Arbitrary geometric layouts disrupt the regular structure required for virtual array construction, limiting the performance of methods based on differential comatrix and spatial smoothing. Meanwhile, subspace-based methods such as the MUSIC and ESPRIT algorithms rely on accurate estimation of the sample covariance matrix. Under conditions of limited snapshots or low signal-to-noise ratio, the signal and noise subspaces are difficult to separate effectively, leading to decreased resolution and increased estimation errors. Mesh-based sparse reconstruction methods are significantly affected by off-mesh bias. While denser meshes can mitigate such errors, computational and storage costs increase dramatically. Existing off-mesh correction or semi-meshless methods often lack stability when dealing with multi-source, coherent signals or single-snapshot scenarios. While high-precision schemes such as sparse Bayesian learning and atomic norm minimization can improve estimation performance, the large-scale matrix inversion or semidefinite programming problems they involve introduce high computational complexity, making them unsuitable for real-time processing. Therefore, in response to practical challenges such as non-integer arrays, limited snapshots, and low signal-to-noise ratios, developing robust and efficient DOA estimation methods has become a critical issue that urgently needs to be explored. Summary of the Invention

[0005] In view of the above-mentioned deficiencies of the prior art, this invention provides a non-integer linear array DOA estimation method based on accelerated iterative hard thresholding. This method expands the array degrees of freedom by constructing a differential comatrix, uses the covariance vector of a single snapshot sample as the sparse representation input, combines an iterative hard thresholding algorithm and the Armijo step size criterion to adaptively accelerate convergence, and compensates for off-grid errors caused by mesh partitioning using the total least squares method of Taylor expansion, thereby achieving high-precision DOA estimation with a small number of snapshots.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] The method for estimating the DOA of non-integer linear arrays based on accelerated iterative hard thresholding includes the following steps:

[0008] Step 1: Construct a non-integer linear array comprising multiple sensors and its received signal model; generate an estimate of the ideal covariance matrix of the received signal model using a finite number of snapshots;

[0009] Step 2: Based on the received signal model, construct the differential co-matrix and sparse optimization model of the non-integer linear array;

[0010] Step 3: Using the iterative hard thresholding algorithm, combined with the Armijo-type step size criterion to automatically adjust the step size and support set, solve the sparse optimization model to obtain the initial angle estimate;

[0011] Step 4: Generate an accurate angle estimate by performing a first-order Taylor expansion on the initial angle estimate.

[0012] Preferably, step 1 includes:

[0013] Step 11: Construct the non-integer linear array and obtain the steering matrix of the non-integer linear array; Step 12: Construct the received signal model based on the steering matrix;

[0014] Step 13: Define an ideal covariance matrix based on the received signal model; generate the estimated value of the ideal covariance matrix using a finite number of snapshots.

[0015] Preferably, step 2 includes:

[0016] Step 21: Construct a difference comatrix of the non-integer linear array based on the element differences of the sensor positions in the non-integer linear array; sort the elements in the difference comatrix in order.

[0017] Step 22: Construct the signal-of-arrival (SOA) model by vectorizing the estimated values;

[0018] Step 23: The received signal model is transformed into a sparse representation model by dividing the range of the direction of arrival into multiple uniformly distributed discrete directions and forming a discretized grid.

[0019] Step 24: Perform a real and imaginary part separation operation on the sparse representation model to generate a real sparse coefficient vector; then, by applying sparsity constraints to the real sparse coefficient vector, generate the sparse optimization model.

[0020] Preferably, step 22 includes:

[0021] The steering vector of the differential comatrix is ​​obtained, and then the differential comatrix steering matrix is ​​generated. Combined with the vectorized estimated value, the signal-to-arrival model is constructed.

[0022] Preferably, step 3 includes:

[0023] Iteratively update the real-valued sparse coefficient vector; calculate the gradient of the sparse optimization model; set the update rule; select a support set containing 2K indices. ;

[0024] The update rule is optimized by introducing a sub-optimization problem that minimizes the sparse optimization model;

[0025] During gradient descent, an Armijo line is used to search for the step size rule; after terminating the iteration, the real and imaginary parts of the updated real sparse coefficient vector are added element-wise to obtain a complex vector; the complex vector is then moduloed, and the maximum value is used to determine the step size. The initial angle estimate is obtained by combining the positions of each element with the sampling grid.

[0026] Preferably, the Armijo line search step size rule in step 3 includes:

[0027] Assume the first The real sparse coefficient vector is obtained during the next iteration. The estimated value Selected Set the initial step size; select The scaling parameter is used to iteratively update the real sparse coefficient vector using the following formula. The estimated value :

[0028]

[0029] in, , It is the smallest non-negative integer that satisfies the following conditions:

[0030]

[0031] Based on the real sparse coefficient vector The estimated value calculate gradient Sparse optimization model At that time, the guiding matrix after performing the real and imaginary part separation operation on the sparse optimization model is... and real sparse coefficient vector The estimated value Limited to its support set Instead of calculating everything, reduce the amount of computation.

[0032] ,

[0033] Subscript express and Only include its index belonging to The element, if in the first... The maximum number of iterations was reached in the next iteration. Or satisfy Then the iteration terminates.

[0034] Preferably, step 4 includes:

[0035] The offset is calculated by performing a Taylor expansion on the initial angle estimate;

[0036] Steering vector of the difference comatrix Performing the first-order Taylor expansion, we obtain:

[0037]

[0038] in This indicates the estimation based on the array manifold of the differential comatrix and the initial angle. The constructed guidance matrix is ​​defined. To account for the off-grid error in angle estimation, the array output is rewritten as:

[0039]

[0040] in , ;

[0041] make The solution to the above equation is obtained by performing the total least squares method:

[0042]

[0043] The elements contain precise angle estimates. and the initial estimated angle Off-network error The precise angle estimate is generated as follows:

[0044] .

[0045] Compared with the prior art, the present invention has the following advantages:

[0046] 1. This invention combines differential co-matrix with sparse representation, employs an iterative hard thresholding algorithm for fast sparse reconstruction, innovatively introduces the Armijo step size criterion to accelerate convergence, and restricts the gradient descent steps to the support set, thereby significantly reducing computational complexity. Furthermore, it utilizes Taylor expansion's total least squares method to compensate for coarse grid off-grid errors, thus significantly improving the DOA estimation accuracy of non-integer arrays.

[0047] 2. Unlike existing compressed sensing gradient descent schemes that use fixed or empirical step sizes, this invention employs the Armijo step size criterion. This allows for adaptive determination of the step size in each iteration, ensuring sufficient descent, resulting in more robust convergence and fewer iterations. This facilitates stable operation under conditions of low SNR and time constraints. This invention introduces the Armijo step size criterion to adaptively select the gradient descent step size, ensuring sufficient descent of the objective function and improving convergence speed. During Armijo line search, the main computations are restricted to the support set, significantly reducing computational complexity.

[0048] 3. Unlike existing schemes that calculate gradients and objective functions on a complete dictionary or guiding matrix, this invention limits the calculation of gradients and objective functions to the current support set, enabling small-scale matrix and vector operations to be performed in a single iteration, thereby significantly reducing the amount of computation and facilitating efficient implementation.

[0049] 4. Unlike schemes that rely solely on sparse reconstruction or single-step off-net correction, this invention combines differential co-array sparse representation, iterative hard threshold reconstruction, and total least squares off-net correction based on first-order Taylor expansion to form a holistic optimization framework for non-integer linear arrays. This scheme achieves sparse recovery in the co-array domain to expand the equivalent aperture, and then performs local offset correction near the initial angle, satisfying the accuracy requirements of global sparsity and local fine-grained correction. By introducing optimization techniques such as adaptive step size and support set constraints into the algorithm structure, convergence stability and noise resistance are improved. Thus, under conditions of non-integer array layout and snapshot constraints, it overcomes the problems of virtual aperture destruction and off-net error accumulation in traditional methods, achieving higher estimation accuracy and robustness. Attached Figure Description

[0050] Figure 1 This is a schematic diagram of the method flow of Embodiment 1 of the present invention;

[0051] Figure 2 This is a schematic diagram of RMSE under different SNR conditions in Embodiment 2 of the present invention;

[0052] Figure 3 This is a schematic diagram of RMSE under different array aperture conditions in Embodiment 2 of the present invention;

[0053] Figure 4 This is the DOA estimation result of 50 Monte Carlo trials with a snapshot number T=30 according to the method of the present invention in Example 2;

[0054] Figure 5 This is the DOA estimation result of 50 Monte Carlo trials with a snapshot number T=30 using the MLE method in Example 2;

[0055] Figure 6 This is the DOA estimation result of 50 Monte Carlo trials with a snapshot number T=30 using the MUSIC method in Example 2;

[0056] Figure 7 This is the DOA estimation result of 50 Monte Carlo trials with a snapshot number T=300 according to Example 2;

[0057] Figure 8 This is the DOA estimation result of 50 Monte Carlo trials with a snapshot number T=300 using the MLE method in Example 2;

[0058] Figure 9This is the DOA estimation result of 50 Monte Carlo trials with a snapshot number T=300 using the MUSIC method in Example 2;

[0059] Figure 10 This is a schematic diagram of the random selection of array element positions in Embodiment 2 of the present invention;

[0060] Figure 11 This is a schematic diagram of the initial DOA estimation for 13 incident sources in Embodiment 2 of the present invention;

[0061] Figure 12 This is the objective function of Embodiment 2 of the present invention. A schematic diagram of the convergence curve with the number of iterations. Detailed Implementation

[0062] To make the technical means, inventive features, objectives, and effects of the invention readily understandable, the invention is further described below with reference to specific illustrations. However, the invention is not limited to the embodiments described below.

[0063] It should be understood that the structures, proportions, sizes, etc., shown in the accompanying drawings of this specification are only used to complement the content disclosed in the specification for those skilled in the art to understand and read, and are not intended to limit the conditions under which the present invention can be implemented. Therefore, they have no substantial technical significance. Any modifications to the structure, changes in the proportions, or adjustments to the size, without affecting the effects and objectives that the present invention can produce, should still fall within the scope of the technical content disclosed in the present invention.

[0064] This invention is based on sparse representation, employs iterative hard threshold optimization and introduces the Armijo step size criterion to accelerate convergence, and combines off-network error compensation to improve estimation accuracy.

[0065] Example 1:

[0066] like Figure 1 The method shown here is a non-integer linear array DOA estimation method based on accelerated iterative hard thresholding, which includes the following steps:

[0067] Step 1: Construct a non-integer linear array comprising multiple sensors and its received signal model; generate an estimate of the ideal covariance matrix of the received signal model using a finite number of snapshots;

[0068] Step 11: Construct a non-integer linear array and obtain the guiding matrix of the non-integer linear array;

[0069] Build a - An arbitrary non-integer linear array of array elements, whose sensor position distribution is as follows:

[0070]

[0071] in It is the unit spacing. For wavelength, , It is the first The position of each array element, and It is a non-integer. Assuming the number of sources is known, we have The incident angles of the three uncorrelated far-field narrowband signals are: , The guiding matrix of the array for:

[0072]

[0073] in Represents the set of complex numbers, corresponding to the direction. The steering vector is:

[0074]

[0075] in It is a natural constant. It is the imaginary unit.

[0076] Step 12: Construct the received signal based on the steering matrix;

[0077] Array received signal The model is as follows:

[0078] ,

[0079] in It is a signal vector. It has zero mean and variance. Complex Gaussian noise, Indicates the number of snapshots.

[0080] Step 13: Define the ideal covariance matrix based on the received signal; generate an estimate of the ideal covariance matrix using a finite number of snapshots.

[0081] Assume that the different signals and noise are statistically independent of each other. The ideal covariance matrix of the received signal is defined as:

[0082]

[0083] in Representing the The power of each signal, The mathematical expectation operator, express Identity matrix. In practice, The estimated value is obtained using a limited number of snapshots.

[0084]

[0085] From the above equation, we can observe the matrix. For all All have elements That is, it depends only on the difference regarding the physical sensor location. .

[0086] Step 2: Based on the received signal model, construct the differential co-matrix and sparse optimization model of the non-integer linear array;

[0087] Step 21: Construct a difference comatrix of the non-integer linear array based on the element differences of the sensor positions in the non-integer linear array; sort the elements in the difference comatrix in order.

[0088]

[0089] It can be viewed as a virtual array, whose sensor positions are determined by... The differences between the elements in the expression are given.

[0090] Difference coarray After sorting the elements in the array, it is re-represented as:

[0091]

[0092] in , The position of the virtual array element in the array. The number of array elements.

[0093] Step 22: Construct the signal-of-arrival (SOA) model using vectorized estimates;

[0094] Vectorized With the formation Received equivalent signal There is a correspondence between them because both depend on the vectorized result of the difference between the actual array element positions:

[0095]

[0096] in and These represent vectorization, Kronecker product, and conjugate operations, respectively. .therefore This can be viewed as single-snapshot data received on an equivalent co-array of a non-integer linear array. Let... The following signal model is obtained:

[0097]

[0098] in

[0099] ,

[0100]

[0101] It can be considered as a co-array guiding matrix, where the guiding vector is

[0102]

[0103] Step 23: Convert the signal-of-arrival (SOA) model into a sparse representation model;

[0104] To transform the Direction of Arrival (DOA) estimation problem into a sparse representation problem, we assume... To uniformly cover the range of the wave direction of arrival The discretized grid, where The number of grid cells. If the grid is fine enough that the actual DOA lies on (or is practically close to) the grid, it is possible to... Use the following model:

[0105]

[0106] in, The guiding matrix is ​​constructed based on the discretized grid. and , For the sparse coefficient vector that needs to be estimated, The The row corresponds to from The signal from the possible source impacting the array, For noise, we have observation noise and model error caused by the discrete grid approximating the continuous direction.

[0107] Step 24: For the sparse representation model Perform the operation of separating the real and imaginary parts:

[0108]

[0109] For complex models Performing a real-to-imaginary part separation operation yields a sparse representation model of real numbers. .

[0110] in, and Let represent the real and imaginary parts, respectively. This is achieved by using a sparse vector of real coefficients. Apply Sparsity constraints lead to the following sparse optimization model:

[0111]

[0112] express Norm. Therefore, the DOA estimation problem for the non-integer linear array NILA is transformed into solving a sparse optimization model.

[0113] Step 3: Using the iterative hard thresholding algorithm, combined with the Armijo-type step size criterion to automatically adjust the step size and support set, solve the sparse optimization model to obtain the initial angle estimate;

[0114] Iterative hard thresholding algorithms have attracted widespread attention due to their excellent recovery properties. This invention utilizes the framework of iterative hard thresholding algorithms, combined with an Armijo-type step-size criterion to automatically adjust the step size and support set to solve sparse optimization models. The main steps are as follows: [The text abruptly ends here, likely due to an incomplete sentence or missing information.] Iterative updates will be performed.

[0115] First calculate The gradient is used for gradient descent, i.e.

[0116]

[0117] Set update rules:

[0118]

[0119] in This is the iteration step size. (Choose...) for Best Support. That is, It contains a set of indexes that define The largest Elements with absolute values.

[0120] Selected back, The steps will Solving the optimization problem within a subspace can provide better estimates and reduce computational complexity.

[0121]

[0122] This subspace is reserved only The index belongs to It is composed of elements, and all other elements are set to zero.

[0123] For step size In order to make the function Minimizing it as much as possible introduces a sub-optimization problem:

[0124]

[0125] If precise line search is used to find the precise optimal step size Exact line search increases the complexity of the optimization problem and incurs significant computational costs, thus it is rarely used in practical applications. This invention chooses to use the classic Armijo line search step size rule during gradient descent. The core ideas of the Armijo criterion are twofold: 1) the objective function value should decrease sufficiently; 2) the search step size... It shouldn't be too small. Assuming the... During the next iteration, we obtain The estimated value Select For the initial step size and The scaling parameter is calculated iteratively. ,in , It is the smallest non-negative integer that satisfies the following conditions

[0126]

[0127] parameter For a very small positive number, this step ensures that from arrive The reduction is significant.

[0128] The above formula makes the first -th iteration The number of non-zero elements in the array does not exceed Therefore, the calculation... and At that time, it can be Limited to its support set Reduce the amount of computation, rather than calculating everything.

[0129] ,

[0130] Subscript express and Only include its index belonging to The element. If in the first The maximum number of iterations was reached in the next iteration. Or satisfy Then the iteration terminates.

[0131] Finally, the real and imaginary parts are added element by element, i.e. , to obtain complex vector .right Take the mold According to the maximum The initial angle estimate can be obtained by combining the position of each element with the sampling grid. .

[0132] Step 4: Generate an accurate angle estimate by performing a first-order Taylor expansion on the initial angle estimate.

[0133] In reality, the true DOA is rarely precisely located on a pre-defined spatial grid, requiring further off-grid error correction of the initial estimate. This is mainly achieved by performing a Taylor expansion on the initial estimate to calculate the offset.

[0134] Steering vector for virtual array Performing the first-order Taylor expansion, we can obtain...

[0135]

[0136] in This indicates that the estimation is based on the array manifold and initial angle of the virtual array. The constructed guidance matrix is ​​defined. The off-grid error for angle estimation can be rewritten as...

[0137]

[0138] in , .make The solution to the above equation can be obtained by performing the total least squares method.

[0139]

[0140] The elements contain precise angle estimates. and initial estimated angle Off-network error Therefore, the final precise estimate can be written as:

[0141]

[0142] The main calculation in this invention involves obtaining an initial angle estimate using an iterative hard thresholding method. Each iteration involves Armijo line search and gradient calculation, with a computational complexity of approximately O(n log n). ,in This is the number of iterations for the Armijo line search, typically set to a maximum of 15.

[0143] Example 2:

[0144] This section uses numerical simulations to investigate the performance of the proposed method. The invention is compared with the methods described below:

[0145] 1) Phase-Difference Projection (PDP)

[0146] Algorithm [DOA Estimation With Non-Uniform Linear Arrays:

[0147] A Phase-Difference Projection Approach];

[0148] 2) Two-step Off-grid DOA Estimation

[0149] with Arbitrary-Spaced Linear Array using Single Snapshot];

[0150] 3) Multiple emitter location and signal parameter estimation

[0151] estimation,MUSIC)[Multiple emitter location and signal parameter estimation];

[0152] 4) Maximum Likelihood Estimator (MLE)

[0153] [Threshold region performance of maximum likelihood direction of

[0154] [arrival estimators]

[0155] 5) Cramér–Rao Lower Bound (CRLB)

[0156] [Threshold region performance of maximum likelihood direction of

[0157] [arrival estimators]

[0158] For the MUSIC and MLE algorithms, use Spectral peak searches were performed at intervals. The root mean square error (RMSE) was used to verify the accuracy of the DOA estimation.

[0159]

[0160] in Denotes the incident angle of the nth Monte Carlo experiment. The estimated value was obtained. The simulation was performed in Matlab R2020B on a system with an Intel Core i5-10400 CPU and 16GB of RAM.

[0161] like Figure 2 , 3 As shown, this invention verifies that all methods use a single snapshot ( Estimate the capability of a single incident source. To avoid the incident angle falling exactly on the discrete space grid, the incident angle is set to [value missing] for each simulation. ,in obey The array elements are uniformly distributed. The number of array elements is fixed. The number of Monte Carlo experiments was 1000. For each Monte Carlo experiment, and the position of the array element All are regenerated.

[0162] The RMSE variation curves of all methods with input signal-to-noise ratio (SNR) are as follows: Figure 2 As shown, the aperture of the non-integer linear array is fixed at... This invention demonstrates that when the input SNR is low, all methods perform poorly due to the noise 'drowning' useful information in the received data, resulting in an RMSE significantly higher than CRLB. As the input SNR increases, the RMSE of this invention decreases rapidly, achieving performance close to CRLB and outperforming all other comparative methods. Under high input SNR conditions, methods such as PDP, MLE, and MUSIC all achieve good estimation performance, while the Two-Step method has a higher RMSE than the others.

[0163] Figure 3 The RMSE of the present invention, PDP, MLE, and MUSIC methods as a function of aperture length was plotted. The curve showing the change in SNR, where SNR is fixed at 10dB. Because... Figure 1The two-step method performs poorly at high SNR levels, therefore its results are not presented. It can be observed that, with a fixed number of elements, the RMSE and CRLB of different methods decrease as the array aperture increases. The RMSE of this invention remains close to the CRLB and is lower than the other comparative methods.

[0164] Because the simulation uses a fixed aperture length This invention employs randomly generated array element positions. In practical applications, the array elements can be arranged according to actual needs. Furthermore, existing research has shown that the output of any sensor is affected by its adjacent elements, i.e., mutual coupling between array elements, and the magnitude of the coupling coefficient is inversely proportional to the distance between the elements. This invention allows for the flexible arrangement of a small number of array elements within a given large array aperture, and the larger element spacing helps to reduce the impact of mutual coupling effects between elements.

[0165] like Figure 10 The ability of this invention to distinguish multiple similar sources was verified. Since the PDP method is applicable to a single snapshot and the estimation of a single incident source, this invention only compares its estimation results with those of the MLE and MUSIC methods. The simulation considered four unrelated sources from different angles. Impacting a non-integer linear array, obey Uniform distribution, array aperture is The number of array elements is Quick shot number Each Monte Carlo experiment Both the array element positions and the array element positions are regenerated.

[0166] Figure 4 , Figure 5 , Figure 6 The results of 50 Monte Carlo trials are presented. The blue asterisks (*) mark the true incident angles, and the red "o" marks the estimated DOA values. It can be observed that this invention accurately distinguishes four adjacent incident sources in each experiment, while the MLE method shows a few instances where the DOA estimates deviate significantly from the true values. However, the MUSIC method exhibits a large number of estimation failures. The RMSE values ​​of the three methods are 0.1691, 16.8364, and 1.8285, respectively, with this invention achieving the lowest RMSE.

[0167] Since subspace methods typically require a large number of snapshots to accurately construct the signal and / or noise subspaces, this invention aims to further verify the DOA estimation performance of different methods when the number of snapshots is large. Figure 7 , Figure 8 , Figure 9The estimation results of the three methods were compared when the number of snapshots T=300. It can be observed that both the present invention and the MLE method can accurately distinguish the four incident sources, while the MUSIC method still has a relatively large number of estimation failures. The RMSE of the three methods are 0.0588, 12.4227 and 0.2360, respectively, all of which decrease with the increase of the number of snapshots, while the present invention achieves the best estimation accuracy.

[0168] Figure 11 This verifies the invention's ability to identify more sources than the number of physical sensors. Consider 13 incident sources impacting INLA, with incident angles of... k=1,2,…,13, and the number of snapshots is set to 500. The positions of the non-integer linear array elements are determined from the aperture. Ten positions are randomly selected from the data, such as... Figure 4 As shown. Figure 5 The initial estimation results of the present invention are shown, and it can be observed that the present invention can distinguish more sources than the number of sensors, and has the ability to resolve a large number of sources.

[0169] Figure 12 This invention demonstrates the objective function for estimating the initial values ​​of a large-scale incident source. The convergence curve varies with the number of iterations. It can be observed that when applying the iterative hard thresholding algorithm combined with the Armijo-type step size criterion for iterative optimization, It tends to stabilize in less than 30 iterations, exhibiting fast convergence.

Claims

1. A non-integer linear array DOA estimation method based on accelerated iterative hard thresholding, characterized in that, The method comprises the following steps: Step 1, constructing a non-integer linear array comprising a plurality of sensors and a received signal model thereof; generating an estimated value of an ideal covariance matrix of the received signal model by a limited number of snapshots; Step 2, constructing a difference co-array and a sparse optimization model of the non-integer linear array according to the received signal model; comprising: Step 21, constructing a difference co-array of the non-integer linear array according to element difference of sensor positions in the non-integer linear array; sequentially ordering elements in the difference co-array; Step 22, constructing a wave arrival signal model by vectorizing the estimated value; Step 23, converting the received signal model into a sparse representation model by dividing a range of a wave arrival direction into a plurality of uniformly distributed discrete directions and forming a discretization grid; Step 24, performing real and imaginary separation operation on the sparse representation model to generate a real sparse coefficient vector; and then generating the sparse optimization model by applying a sparsity constraint on the real sparse coefficient vector; Step 3, solving the sparse optimization model by using an iterative hard threshold algorithm, combining an Armijo type step length criterion to automatically adjust a step length and a support set, and obtaining an initial angle estimation; comprising: iteratively updating the real sparse coefficient vector; computing a gradient of the sparse optimization model; setting an update rule; selecting a support set containing 2K indices ; Optimizing the update rule by introducing a sub-optimization problem of minimizing the sparse optimization model; Armijo line search step size rule is used in the gradient descent process; after the iteration is terminated, the real parts and the imaginary parts of the updated real sparse coefficient vector are added element by element to obtain a complex vector; the complex vector is taken modulo to obtain the initial angle estimation according to the positions of the maximum elements in combination with a sampling grid Step 4, generating an accurate angle estimation by performing first-order Taylor expansion on the initial angle estimation.

2. The method of claim 1, wherein, Step 1 comprises: Step 11, constructing the non-integer linear array and obtaining a steering matrix of the non-integer linear array; Step 12, constructing the received signal model according to the steering matrix; Step 13, defining an ideal covariance matrix according to the received signal model; and generating the estimated value of the ideal covariance matrix by a limited number of snapshots.

3. The method of claim 1, wherein, Step 22 comprises: Obtaining steering vectors of the difference co-array, and then generating a difference co-array steering matrix; and constructing the wave arrival signal model by vectorizing the estimated value.

4. The method of claim 1, wherein, The Armijo line search step length rule in step 3 comprises: Assume the first The real sparse coefficient vector is obtained during the next iteration. The estimated value Selected Set the initial step size; select The scaling parameter is used to iteratively update the real sparse coefficient vector using the following formula. The estimated value : ； wherein , is the smallest non-negative integer satisfying: ； Based on the real sparse coefficient vector The estimated value calculate gradient Sparse optimization model At that time, the guiding matrix after performing the real and imaginary part separation operation on the sparse optimization model is... and real sparse coefficient vector The estimated value Limited to its support set Instead of calculating everything, reduce the amount of computation. ； ； where the subscript represents and contains only elements whose indices belong to , the iteration is terminated if, on the th iteration, the maximum number of iterations is reached or the condition is met.

5. The acceleration-iteration hard-thresholding based non-integer linear array DOA estimation method according to claim 4, characterized in that, Step 4 comprises: Calculating a deviation by performing Taylor expansion on the initial angle estimation; Steering vector for differential coarrays Performing a first order Taylor expansion, we get: ； wherein denotes the array manifold according to the differential coarray and the initial angle estimate the steering matrix constructed, defining the off-grid error for the angle estimate, the array output is rewritten as: ； wherein , ; Let the solution of the above equation is obtained by performing total least squares ； the elements include a precise estimate of an angle and the off-network error between the initial estimate of the angle generating the precise angle estimate as:​ 。

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